Lemma 85.34.5 (0DHG)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.34: Fppf hypercoverings of algebraic spaces: modules
Lemma 85.34.5 (cite)

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Lemma 85.34.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $U$ be a simplicial algebraic space over $S$ . Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$ , then

$R\Gamma (X_{\acute{e}tale}, K) = R\Gamma (U_{\acute{e}tale}, a^*K)$

for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ . Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 85.32.

Proof. This follows from Lemma 85.34.4 because $R\Gamma (U_{\acute{e}tale}, -) = R\Gamma (X_{\acute{e}tale}, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4. $\square$

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